Harnessing high-dimensional hyperentanglement through
a biphoton frequency comb
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Citation
Xie, Zhenda, Tian Zhong, Sajan Shrestha, XinAn Xu, Junlin
Liang, Yan-Xiao Gong, Joshua C. Bienfang, et al. “Harnessing
High-Dimensional Hyperentanglement through a Biphoton
Frequency Comb.” Nature Photon 9, no. 8 (June 29, 2015):
536–542.
As Published
http://dx.doi.org/10.1038/nphoton.2015.110
Publisher
Nature Publishing Group
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Author's final manuscript
Accessed
Thu May 26 19:40:08 EDT 2016
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http://hdl.handle.net/1721.1/101042
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Harnessing
high-dimensional
hyperentanglement
through
a
biphoton
frequency comb
Zhenda Xie1,2, Tian Zhong3, Sajan Shrestha2, XinAn Xu2, Junlin Liang2,Yan-Xiao Gong4, Joshua
C. Bienfang5, Alessandro Restelli5, Jeffrey H. Shapiro3, Franco N. C. Wong3, and Chee Wei
Wong1,2
1
Mesoscopic Optics and Quantum Electronics Laboratory, University of California, Los Angeles,
CA 90095
2
Optical Nanostructures Laboratory, Columbia University, New York, NY 10027
3
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA
02139
4
Department of Physics, Southeast University, Nanjing, 211189, People’s Republic of China
5
Joint Quantum Institute, University of Maryland and National Institute of Standards and
Technology, Gaithersburg, Maryland 20899, USA
Quantum entanglement is a fundamental resource for secure information processing
and communications, where hyperentanglement or high-dimensional entanglement has
been separately proposed for its high data capacity and error resilience. The
continuous-variable nature of the energy-time entanglement makes it an ideal candidate for
efficient high-dimensional coding with minimal limitations. Here we demonstrate the first
simultaneous high-dimensional hyperentanglement using a biphoton frequency comb to
harness the full potential in both energy and time domain. Long-postulated
Hong-Ou-Mandel quantum revival is exhibited, with up to 19 time-bins and 96.5%
visibilities. We further witness the high-dimensional energy-time entanglement through
Franson revivals, observed periodically at integer time-bins, with 97.8% visibility. This
qudit state is observed to simultaneously violate the generalized Bell inequality by up to
1
10.95 standard deviations while observing recurrent Clauser-Horne-Shimony-Holt
S-parameters up to 2.76. Our biphoton frequency comb provides a platform for
photon-efficient quantum communications towards the ultimate channel capacity through
energy-time-polarization high-dimensional encoding.
Increasing the dimensionality of quantum entanglement is a key enabler for high-capacity
quantum communications and key distribution [1, 2], quantum computation [3] and information
processing [4, 5], imaging [6], and enhanced quantum phase measurement [7, 8]. A large Hilbert
space can be achieved through entanglement in more than one degree of freedom (known as
hyperentanglement [2, 7, 9]), where each degree of freedom can also be expanded to more than
two dimensions (known as high-dimensional entanglement). The high-dimensional entanglement
can be prepared in several physical attributes, for example, in orbital angular momentum [1, 10-12]
and other spatial modes [13-15]. The drawback of these high-dimensional spatial states is
complicated beam-shaping for entanglement generation and detection, which reduces the
brightness of the sources as the dimension scales up, and complicates their use in
optical-fiber-based communications systems. In contrast, the continuous-variable energy-time
entanglement [16-22] is intrinsically suitable for high-dimensional coding and, if successful, can
potentially be generated and be communicated in the telecommunication network. However, most
studies focus on time-bin entanglement, which is discrete-variable entanglement with typical
dimensionality of two [23-25]. Difficulties in pump-pulse shaping and phase control limit the
dimensionality of the time-bin entanglement [26], and high-dimensional time-bin entanglement
has not been fully characterized because of the overwhelmingly complicated analyzing
interferometers. On the other hand, a biphoton state with a comb-like spectrum could potentially
serve for high-dimensional entanglement generation and take full advantage of the
continuous-variable energy-time subspace. Based on this state, promising applications have been
proposed for quantum computing, secure wavelength-division multiplexing, and dense quantum
key distribution [3, 27, 28]. A phase-coherent biphoton frequency comb (BFC) is also known for
2
its mode-locked behavior in its second-order correlation. Unlike classical frequency combs, where
mode-locking directly relies on phase coherence over individual comb lines, the mode-locked
behavior of a BFC is the representation of the phase coherence of a biphoton wavepacket over
comb-line pairs, and results in periodic recurrent correlation at different time-bins [29, 30]. This
time correlation feature can be characterized through quantum interference when passing the BFC
through an unbalanced Hong-Ou-Mandel (HOM)-type interferometer [31]. A surprising revival of
the correlation dips can be observed at time-bins with half the period of the BFC revival time.
However, because of the limited type-I collinear spontaneous parametric downconversion (SPDC)
configuration in the prior studies [29], post-selection was necessary for the BFC generation where
the signal and idler photons are indistinguishable, limiting the maximum two-photon interference
to 50 %.
Here we achieve high-dimensional hyperentanglement through a biphoton frequency comb.
The high-dimensional hyperentanglement of the BFC is fully characterized with four observations.
First, the state is prepared at telecommunications wavelengths, without the necessity of
post-selection, by using a type-II high-efficiency periodically-poled KTiOPO4 (ppKTP)
waveguide together with a fiber Fabry-Perot cavity (FFPC). Because of the type-II phase matching,
signal and idler photons can be separated efficiently by a polarizing beamsplitter (PBS), allowing
deterministic BFC generation, as first proposed theoretically by one of the authors [29], to be
observed experimentally for the first time. Revival dips with  96.5 % visibility from two-photon
interference in a HOM-type interferometer are observed recurrently for the first time, which
reveals correlation features in the time bins of the BFC. Second, second-order frequency
correlation and anticorrelation, scanned across the full span of the frequency bins by narrowband
filter pairs, shows the good fidelity of the frequency-bin entangled state. Finally, we confirm the
generation of high-dimensional energy-time entanglement using a Franson interferometer, which
can be regarded as a generalized Bell inequality test. For the first time, Franson interference
fringes are observed to revive periodically at different time-bin intervals with visibilities up to
3
97.8 %. Based on these three measurements, we encode extra qubits in the BFC photon pairs by
mixing them on a 50:50 fiber beam splitter (FBS). We witness the hyperentanglement by
simultaneous polarization and Franson interferometer analysis, demonstrating a generalized Bell
inequality in two polarization and four time-bin subspaces up to 10.95 and 8.34 standard
deviations respectively. The Clauser-Horne-Shimony-Holt (CHSH) S-parameters are determined
for the polarization basis across the different time-bins with a maximum up to 2.76.
Figure 1(a) shows our experimental scheme. The SPDC entangled photon pairs are generated
by a high-efficiency type-II ppKTP waveguide, described in detail in [32]. The
frequency-degenerate SPDC phase matching is designed for 1316 nm output with a bandwidth of
about 245 GHz. The type-II BFC is generated by passing the SPDC photons through a fiber
Fabry-Perot cavity (Micron Opticsi) with the signal and idler photons in orthogonal polarizations
(H: horizontal and V: vertical). A BFC state is expressed as:
 
N
  d  f (  m)aˆ
m  N
†
H
( p / 2  )aˆV† ( p / 2  ) 0 ,
(1)
where  is the spacing between the frequency bins, i.e., the free spectral range of the FFPC in
rad/s;  is the detuning of the SPDC biphotons from their central frequency; and the state’s
spectral amplitude, f (  m) , is the single frequency bin profile, defined by the
Lorentzian-shape transmission of the FFPC with full-width half-maximum (FWHM) of 2 .
f ()  1 [( ) 2   2 ],
(2)
The signal and idler photons are separated with 100 % probability by a polarizing beamsplitter
or, in other words, the BFC is thus prepared without post-selection. The temporal wave function of
the BFC can be written as:
 t   d exp(  )
i
sin[(2 N  1) / 2] †
aˆ H (t )aˆV† (t   ) 0 ,
sin(  / 2)
(3)
The identification of any commercial product or trade name does not imply endorsement or recommendation by the
National Institute of Standards and Technology.
4
where the exponential decay is slowly varying because of small  , and thus the temporal
behavior of the BFC is mainly determined by the term
sin[(2 N  1) / 2]
, with a repetition
sin( / 2)
time T = 2/.
The FFPC has free spectral range (FSR) and bandwidth of 15.15 GHz, 1.36 GHz, respectively.
The repetition period T of the BFC is about 66.2 ps. The FFPC is mounted onto a thermoelectric
cooling sub-assembly with minimized stress to eliminate polarization birefringence and with  1
mK high-performance temperature control. From our measurements, there is no observable
polarization birefringence in the FFPC, and thus the signal and idler photons have the same
spectrum after they pass through the cavity. Due to the type-II configuration, there is no
probability that both photons propagate in the same arm of the HOM interferometer – this
configuration yields a potential maximum of 100 % visibility of the two-photon interference.
The pump is a Fabry-Perot laser diode that is stabilized with self-injection-locking, through
double-pass first-order-diffraction feedback using an external grating (for details, see
Supplementary Information and Methods). It is passively stabilized with modest low-noise current
and temperature control for single-longitudinal-mode lasing at 658 nm, with a stability of less than
2 MHz within 200 s (measured with the Franson interference experiment, see Supplementary
Information). We first match the FFPC wavelength with the pump via temperature tuning and
second harmonic generation (SHG) from a frequency-stabilized distributed feedback (DFB)
reference laser at 1316 nm. The SHG is monitored with a wavemeter (WS-7, HighFinesse) to 60
MHz accuracy, allowing the tuning of the laser diode current and first-order-feedback diffraction
angle to match the FFPC for the BFC generation. The BFC spectrum is further cleaned by a fiber
Bragg grating (FBG) with a circulator, and a free-space long-pass filter that blocks the residual
pump light. The FBG is chosen with a bandwidth of 346 GHz, larger than the 245 GHz
phase-matching bandwidth, and is simultaneously temperature-controlled to match the central
5
wavelength of the SPDC. A polarizing beamsplitter separates the orthogonally polarized signal
and idler photons.
We first characterize the mode-locked behavior using a HOM interferometer. The signal and
idler photons are sent through different arms of the interferometer. A fiber polarization controller
in the interferometer’s lower arm controls the idler photon polarization, so that the two photons
have the same polarization at the FBS. An optical delay line consisting of a prism and a motorized
long-travel linear stage (DDS220, Thorlabs) is used to change the relative timing delay between
the two arms of the HOM interferometer. The position-dependent insertion loss of the optical
delay line is measured to be less than 0.02 dB throughout its entire travel range of 220 mm. The
coincidence measurements are performed with two InGaAs single-photon detectors D1 (with 
20 % detection efficiency and 1 ns gate width) and D2 (with  25 % detection efficiency and 3 ns
gate width). D1 is triggered at 15 MHz, and its output signal is used to trigger D2. As a result,
coincidences can be detected directly from the D2 counting rate if the proper optical delay is
applied to compensate for the electronic delay. Taking into account the waveguide-to-fiber
coupling and transmission efficiencies of optical components, the overall signal and idler detection
efficiencies are estimated to be s = 0.92 % and i = 1.14 % respectively. Figure 1(b) shows the
experimental results by scanning the optical delay between the two arms of the HOM
interferometer from -320 ps to 320 ps, with a pump power of 2 mW. At this power level the
generation rate is about 3.3×10-3 pairs per gate, where multi-pair events can be neglected. We
obtain revival dips for the coincidence counting rate R12 at the two outputs of the HOM
interferometer (single-photon rates are shown in the Supplementary Information). The spacing
between the dips is 33.4 ps, which matches half the repetition period of the BFC, and agrees with
our theory and numerical modeling (see Supplementary Information I for details). We also note
that our prior analysis [30] involved a movable beam splitter for both beams (for a T recurrence)
while in our measurement setup only the idler beam is delayed (for a T/2 recurrence), supported by
the same type of analysis and physical interpretation. The visibility of the dips decreases
6
exponentially [see right insert in Figure 1(b)] due to the Lorentzian lineshape spectra of the SPDC
individual photons after they pass through the FFPC. A zoom-in of the dip around the zero delay
point is shown in the insert of Figure 1(b). The maximum visibility is measured to be 87.2  1.5 %,
or 96.5 % after subtracting the accidental coincidence counts ( 17.3 per 70 sec). Its base-to-base
width is fitted to be 3.86  0.30 ps, corresponding to a two-photon bandwidth of 259  20 GHz,
which agrees with the expected 245 GHz phase-matching bandwidth. More details are provided in
Supplementary Information V. Considering the measured bin spacing /2 of 14.98 GHz, 17
frequency bins are generated in our measurements within the phase matching bandwidth,
equivalent to  4 quantum bits per photon for high-dimensional frequency entanglement.
To test the purity of the high-dimensional frequency bin entanglement, we further measure
the correlation between different frequency bins for the signal and idler photons. Each pair of the
frequency-bins is filtered out by a set of narrow band filters for the signal and idler photons. As
shown in Figure 2, each filter is composed of a FBG centered at 1316 nm with 100 pm FWHM and
an optical circulator. Both FBGs are embedded in a custom-built temperature-controlled housing
for fine spectral tuning. The coincidence counting rate is recorded while the filters are set at
different combinations of the signal-idler frequency bin basis. The tuning range is bounded by the
maximum tuned FBG temperature (up to about 100C currently) and in this case 9 frequency bins
can be examined for the signal and idler biphoton. We denote these bins with numbers #-4 to #4,
with #0 standing for the central bin, as shown in Figure 2(b). We see that high coincidence
counting rates are measured only when the filters are set at the corresponding positive-negative
frequency bins, according to Equation 1. The suppression ratio between the corresponding and
non-corresponding counting rate exceeded 13.8 dB for adjacent bins (including  2 % leakage
from the band pass filters), or 16.6 dB for the non-adjacent bins. We note that, through conjugate
state projection [20], observation of both the HOM recurrence and frequency correlation supports
the entangled nature of the BFC.
7
To further characterize the energy-time entanglement of the BFC, we build an interferometer
to examine Franson interference as schematically shown in Figure 3(a). The Franson
interferometer [16, 33] comprises two unbalanced Mach-Zehnder interferometers (detailed in
Methods and Supplementary Information III), with imbalances T1 and T2. Usually, Franson
interference comes from two indistinguishable two-photon events: both signal and idler photons
take the long arm (L-L) and both photons take the short arm (S-S). This interference only happens
when T  T1 - T2 = 0 to within the single photon coherence length, which is 1.81 ps in our case.
Because of the high-dimensional frequency entanglement, however, Franson interference can be
observed at different time revivals. As discussed before, the BFC features a mode-locked revival
in temporal correlation, with repetition period of T. Therefore such Franson interference revival
can be observed for integer temporal delays T = NT where N is an integer including 0. Detailed
theory and modeling are illustrated in Supplementary Information II for the revival of the Franson
interference. The experimental setup is shown in Figure 3(b). The signal and idler photons are sent
to two fiber-based interferometers (arm1 and arm2) with imbalances T1 and T2, respectively.
Details on the pump laser and interferometer setup stabilization are described in Supplementary
Information III and IV, and in the Methods. Each arm is formed by double-pass
temperature-stabilized Michelson interferometers, and two of the output ports of the fiber 50:50
beamsplitter are spliced onto two Faraday mirrors, with single photons being collected from the
reflection. Thus they work effectively as a Mach-Zehnder interferometer, and the polarization
instability inside the fibers is accurately self-compensated. Following the general requirement for
Franson interference, the T1 and T2 time differences (5 ns in our setup) are tuned to be much
larger than the single-photon coherence time and the timing jitter of the single-photon detectors.
The relative timing between the single photon detection gates are programmed such that events are
only recorded for the L-L and S-S events. Both arms are mounted on aluminum housings with  1
mK temperature control accuracy. Enclosures are further used to seal the interferometers with
additional temperature control for isolating mechanical and thermal shocks. To detect the Franson
8
interference at different temporal revivals, we include a free-space delay line on arm2, which gives
us the capability to study the Franson interference with possible detuning T up to 360 ps, bounded
by the free-space delay line travel and for up to six positive-delay time bins.
Here we are interested in the regime of T  0, because of the symmetry of the signal and idler.
In particular, this allows us to examine the interference visibility over a larger time frame, for the
same stage travel range. With T2 fixed at each point, the phase sensitive interference is achieved
by fine-control of the arm1 temperature, which tunes T1. As shown in Figure 4, the revival of
Franson interference is only observed exactly at the periodic time bins T = NT, while no
interference is observed for other T values between these bins. The period of the revival time bins
is 66.7 ps, which corresponds exactly to the round-trip time of the FFPC (2/). The visibility of
the interference fringes are measured to be 94.2 %, 89.3 %, 79.7 %, 71.1 %, 51.0 % and 43.6 %, or,
after subtracting the accidental coincidence counts, 97.8 %, 93.3 %, 83.0 %, 74.1 %, 59.0 %, and
45.4 % respectively. The visibility decreases because of the non-zero linewidth of each frequency
bin and is captured in our theory (see Supplementary Information II). We understand this as a
generalized Bell inequality violation for a high-dimensional state at 4 time bins from the center.
We only measured for T  0, but could expect the violation of the generalized Bell inequality at
the other three inverse symmetric time bins of T at -66.7 ps, -133.4 ps, and -200.1 ps based on
symmetry with the three positive time bins observed.
Figure 5 next shows the high-dimensional hyperentanglement by mixing the signal and idler
photons on a 50:50 FBS (detailed in Supplementary Information VI). In the experiment,
hyperentanglement is generated based on the HOM-interference setup fixed at the central dip, but
with polarization adjusted so that the signal and idler photons are orthogonally polarized at the
FBS for hyperentanglement generation. Two polarizers P1 and P2 are used for the polarization
analysis, which is cascaded with the Franson interferometer for the energy time entanglement
measurement at the same time. One set of half and quarter waveplates are placed before each
polarizer to compensate the polarization change in the fiber after the FBS. An additional thick
9
multi-order full waveplate is used in the lower arm and it can be twisted along its fast axis (fixed in
the horizontal plane) to change the phase delay between the horizontally and vertically polarized
light. Therefore, we successfully generated the following high-dimensional hyperentangled state
when a coincidence is measured.
 
N
  d  f (  m)( H , 
m  N
p
/ 2
1
V ,p / 2  
2
 V ,p / 2  
1
H ,p / 2   )
(4)
2
Such hyperentanglement is tested using coincidence measurement by scanning P2 and the Franson
interferometer while P1 is set at 45° or 90° and T fixed at time bins #0 to #5. A 1.3 GHz
self-differencing InGaAs single-photon detector [34, 35] is used as D1 to maximize the gated
detection rate. As shown in Figure 5(b) and 5(c), we obtained clear interference fringes over both
polarization and energy-time basis, with visibilities up to 96.7 % and 95.9 % respectively, after
dark count subtraction of a 0.44 s-1 rate. Analyzed within the two polarization and four time-bin
subspaces, this corresponds to Bell violation up to 10.95 and 8.34 standard deviations, respectively.
The CHSH S parameter is observed for the polarization basis up to S = 2.76, with details on the
photon statistics shown in Figure 5(d) and listed in Table 1 of the Supplementary Information VII.
In summary, we have demonstrated a high-dimensional hyperentanglement of polarization
and energy-time subspaces using a BFC. Based on continuous-variable energy-time entanglement,
unlimited qubits can be coded on the BFC by increasing the number of comb line pairs with plane
wave pump. Here we show an example for bright BFC generation with 5 qubits per photon, which
is from a type-II SPDC process in a ppKTP waveguide without post-selection. 19
Hong-Ou-Mandel dip recurrences with a maximum of 96.5 % visibility in a stabilized
interferometer are observed. High-dimensional energy-time entanglement is proven by Franson
interference, which can be considered as a generalized Bell inequality test. Revival of the Franson
interference has been witnessed at the periodic time bins, where the time bin separation is the
cavity round trip time, i.e., the revival time of the BFC. The interference visibility is measured up
to 97.8 %. The generalized Bell inequality has been violated at 4 out of 6 measured time bins, or 7
time bins in total considering the symmetry. High-dimension hyperentanglement is further
10
generated and characterized with high fidelity in both polarization and energy-time subspaces,
with Bell inequality violations up to 10.95 and 8.34 standard deviations respectively, with the
measured polarization CHSH S-parameters up to 2.76. It should be noted that the generation rate of
the entangled biphoton frequency comb from our high-performance ppKTP waveguide exceeds
72.2 pairs/s/MHz/mW, by taking account of all detection losses and the 1.5 % duty cycle for the
detection time, which is higher than that of cavity-enhanced SPDC using bulk crystals. If the
propagation loss of the ppKTP waveguide can be further reduced, the waveguide can be put inside
an optical cavity and the brightness of the BFC further enhanced. Such a bright high-dimensional
hyperentangled BFC can encode multiple qubits onto a single photon pair without losing high
photon flux, and thus further increase the photon efficiency with applications in dense quantum
information processing and secure quantum key distribution channels.
Methods
Stabilized self-injection-locked 658 nm pump laser: For the Franson measurements, good longand short-term stability of the 658 nm pump laser is required since this stability defines the
coherence time of the two-photon state. This coherence time should be much longer than the path
length difference ( 5 ns in our Franson measurements). Hence we have designed and built a
stabilized laser through self-injection-locking, in a configuration similar to the Littman-Metcalf
cavity. With its single spatial mode, the longitudinal modes are selected through a diffraction
grating (first order back into the diode; zeroth order as output) with an achieved mode-rejection
ratio of more than 30 dB. Three temperature controllers are used to stabilize the doubly-enclosed
laser system, and a low-noise controller from Vescent Photonics (D2-105) drives the laser diode.
With our design, a 2 MHz stability is achieved for 200 second measurement and integration
timescales, confirmed with the Franson interferometer. For longer timescales (12 hours), a
wavelength meter indicates  100 MHz drift. More details on the characterization, setup, and
design are illustrated in the Supplementary Information.
11
Franson interferometer: For our long-term phase-sensitive interference measurements, the
fiber-based Franson interference needs to be carefully stabilized. Both interferometer arms are
double-enclosed and sealed, and temperature-controlled with Peltier modules. Closed-loop
piezoelectric control fine-tunes the arm2 delay. We have designed and custom-built a pair of fiber
collimators for fine focal adjustment, coupling, and alignment. The double-pass delay line
insertion loss is less than 0.4  0.05 dB over the entire 27 mm delay-travel range (of up to 360 ps in
the reflected double-pass configuration). To verify the Franson interferometer stability, arm1 and
arm2 are connected in series and the interference visibility is observed up to 49.8  1.0 % for both
short and long-term, near or right at the 50 % classical limit. The delay-temperature tuning
transduction is quantified at 127 attoseconds per mK. More details on the characterization, setup,
and design are illustrated in the Supplementary Information.
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14
[30] Shapiro, J. H. Coincidence dips and revivals from a Type-II optical parametric amplifier.
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Acknowledgements
We acknowledge assistance from Todd Pittman, James Franson, and Dov Fields. We also
acknowledge discussions with Abhinav Kumar Vinod, Yongnan Li, Joseph Poekert, Mark Itzler,
Peizhe Li, Dirk R. Englund, and Xiaolong Hu. This work is supported by the InPho program from
Defense
Advanced
Research
Projects
Agency
(DARPA)
under
contract
number
W911NF-10-1-0416. Y.X.G is supported by the National Natural Science Foundations of China
(Grant No. 11474050).
Author contributions
Z.X, T.Z., S.S., X.X., and J.L. performed the measurements, J.C.B. and A.R. developed the 1.3
GHz detectors, T.Z., Y.X.G., F.N.C.W., and J.H.S. provided the theory and samples, and all
authors helped in the manuscript preparation.
15
Additional information
The authors declare no competing financial interests. Supplementary information accompanies
this
paper
online.
Reprints
and
permission
information
is
available
online
at
http://www.nature.com/reprints/. Correspondence and requests for materials should be addressed
to Z.X. (zhenda@seas.ucla.edu) and C.W.W. (cheewei.wong@ucla.edu).
16
Figures
a
BFC source
FBG
ppKTP waveguide
FFPC
Pump laser
Circulator
FPC1
15MHz
clock
PBS
LPF
S.C.
D1
50:50
Coupler
C.C.
500
400
FPC2
400
Experiment
Theory
1.0
200
0
-12
Delay
Visibility
Coincidence Rate (/60s)
b 600
Coincidence Rate (/60s)
D2
0.5
6
0
-6
Relative Delay (ps)
12
0.0
0
200
Relative Delay (ps)
400
300
200
100
0
-300
-200
100
0
-100
Relative Delay (ps)
200
300
Figure 1 | Generation and quantum revival observations of the high-dimensional biphoton
frequency comb. a, Illustrative experimental scheme. FFPC: fiber Fabry-Perot cavity; FBG: fiber
Bragg grating; FPC: fiber polarization controller; LPF: long pass filter; PBS: polarizing
beamsplitter; P: polarizer; S.C.: single counts; C.C.: coincidence counts. b, Coincidence counting
rate as a function of the relative delay T between the two arms of the HOM interferometer. The
HOM revival is observed in the two-photon interference, with dips at 19 time-bins in this case. The
visibility change across the different relative delays arises from the single FFPC bandwidth Δω.
The red solid line is the theoretical prediction from the phase-matching bandwidth. Left insert:
zoom-in coincidence around zero relative delay between the two arms. The dip width was fit to be
17
3.86  0.30 ps, which matches well with the 245 GHz phase-matching bandwidth. The measured
visibility of the dip is observed at 87.2  1.5 %, or 96.5 % after subtracting the accidental
coincidence counts. Right insert: measured bin visibility versus HOM delay, compared with
theoretical predictions (Supplementary Information I).
18
15MHz
clock
FBG1
D1
Circulator
FBG2
-20
-4 -3
-2 -1
0
1
Signa
l freq
uency
b
2
in #
3
0
-1
-2
-3
4 -4
1
#
4
bin
D2
3
fre
que
ncy
Circulator
2
er
C.C.
Idler
-10
Id l
BFC
source
0
Relative coincidenc
Signal
e counting rate (dB)
b
a
Figure 2 | Quantum frequency correlation measurement of the biphoton frequency comb. a,
Experimental schematic for the frequency correlation measurement. Signal and idler photons are
sent to two narrow band filters for the frequency bin correlation measurement with coincidence
counting. Each filter consists of a FBG and a circulator. The FBGs have matched FWHM
bandwidth of 100 pm and are thermally tuned for the scans from -4th to +4th frequency bins from
the center. b, Measured frequency correlation of the BFC. The relative coincidence counting rate is
recorded while the signal and idler filters are set at different frequency-bin numbers.
19
L
T
L
a
T
D1
D2
BFC
source
S
Signal
S
Idler
C.C.
b
Signal
arm1
FRM
Idler
BFC
source
50:50
50:50
T1
FRM
arm2
T2
C.C.
FRM
D1
D2
FRM
15MHz
clock
Figure 3 | Franson interference of the high-dimensional biphoton frequency comb. a,
Schematic map and concept of Franson interference of the BFC. The BFC is prepared with
high-dimensional correlation features of mode-locked behavior with repetition period of T.
Therefore, Franson-type interference between the long-long and short-short events can be
observed when T = NT where N is an integer. b, Experimental Franson interference setup. FRM:
Faraday mirror. The FRMs are used to compensate the stress-induced birefringence of the
single-mode fiber interferometers. A compact optical delay line was used in the longer path of the
arm2 to achieve different imbalances T (=T2 - T1). Both arms are double-temperature
stabilized, first on the custom aluminum plate mountings and second on the sealed enclosures
(light blue thick lines).
20
Coincidence Rate (/200s)
a
a’
800
b
c
d
e
f
200.1
266.9
600
400
200
0
0
3
6
Relative Delay (fs)
12
0
30
66.7
133.4
Time bin (ps)
+a
2.0
+b
h
+c
+d
1.5
+e
a’
+
1.0
+f
b
0.8
-200
-100
0
100
Relative Delay (ps)
c
d
0.6
e
0.4
-300
Experiment
Theory
a
0.5
0.0
333.6
1.0
Visibility
Coincidence Rate, R12
g
9
200
300
0
f
100 200 300
Relative Delay (ps)
400
Figure 4 | Measured Franson interference around different relative delays of arm2. a to f,
Franson interferences at time bin #0 (T = 0), #1 (T = 66.7 ps), #2 (T = 133.4 ps), #3 (T =
200.1 ps), #4 (T = 266.9 ps), and #5 (T = 333.6 ps) respectively. Also included in panel a’ is the
interference measured away from the above time bins at T =30 ps, with no observable
interference fringes. The data points in each panel (of the different relative Franson delays) include
the measured error bars across each data set, arising from Poisson statistics, experimental drift, and
measurement noise. The error bars from repeated coincidence measurements are much smaller
than the observed coincidence rates in our measurement and setup. In each panel, the red line
denotes the numerical modeling of the Franson interference on the high-dimensional quantum
state. g, Theoretical fringe envelope of Franson interference for the high-dimensional biphoton
frequency comb, with superimposed experimental observations. The marked labels (a to f, and a’)
correspond to the actual delay points from which the above measurements are taken. h, Witnessed
visibility of high-dimensional Franson interference fringes as a function of T. The experimental
(and theoretical) witnessed visibilities for the k-th order peaks are 97.8 % (100 %), 93.3 %
(96.0 %), 83.0 % (86.8 %), 74.1 % (75.6 %), 59.0 % (64.0 %), and 45.4 % (53.3 %) respectively.
21
1.3 GHz
clock
S.C.
/2 /4
Signal
50:50
P1
D1
FRM arm1
FRM
BFC
source

FFC2
C.C.
50:50
T = 0
Idler
50:50
FRM arm2
D2
/2 /4  P2
FRM
c
P1=45, pol
P1=45, e-t
P1= -90, pol
P1= -90, e-t
Time Bin #
Coincidence rates (/80s)
Coincidence rates (/80s)
b
d
Bell inequality violation
(standard deviation )
a
Bin 0
Bin 0
Bin 1
Bin 1
Bin 2
P2 angle ()
Bin 3
Bin 2
T (fs)
P2 angle ()
Bin 4
Bin 3
T (fs)
Bin 4
Figure 5 | High-dimensional hyperentanglement on polarization and energy-time basis. a,
Setup for the high-dimensional two degree-of-freedom entanglement measurement. The state is
generated by mixing the signal and idler photons at the 50:50 fiber beam splitter with orthogonal
polarizations. Perfect temporal overlap between signal and idler photons are ensured by the HOM
interference discussed before. High-dimensional hyperentanglement is measured with polarization
analysis using polarizers P1 and P2, cascaded with a Franson interferometer. /2: half wave plate;
/4: quarter wave plate; : multi-order full wave plate. b and c: measured two-photon interference
fringes when P1 is set at 45 and 90 degrees. The period variance of the fringe on the temporal
domain is because of a slow pump laser drift. d, Measured Bell inequality violation at different
time-bins and P1 angles. pol: polarization basis, e-t: energy-time basis.
22
Supplementary Information for
Harnessing
high-dimensional
hyperentanglement
through
a
biphoton
frequency comb
Zhenda Xie1,2, Tian Zhong3, Sajan Shrestha2, XinAn Xu2, Junlin Liang2,Yan-Xiao Gong4, Joshua
C. Bienfang5, Alessandro Restelli5, Jeffrey H. Shapiro 3, Franco N. C. Wong3, and Chee Wei
Wong1,2
1
Mesoscopic Optics and Quantum Electronics Laboratory, University of California, Los Angeles,
CA 90095
2
Optical Nanostructures Laboratory, Columbia University, New York, NY 10027
3
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA
02139
4
5
Department of Physics, Southeast University, Nanjing, 211189, People’s Republic of China
Joint Quantum Institute, University of Maryland and National Institute of Standards and
Technology, Gaithersburg, Maryland 20899, USA
I. Theory of two-photon interference of the high-dimensional biphoton frequency comb
Considering the Hong-Ou-Mandel (HOM) interference at an ideal 50:50 coupler, we can
write the electric field operators at the two detectors D1 and D2 as
1 ˆ
Eˆ1  t  
Es (t )  Eˆ i (t   T )  ,
2
1 ˆ
Eˆ 2  t  
Es (t )  Eˆ i (t   T )  ,
2
(1)
with the field operators before the HOM interferometer Eˆk (t ) , ( k  s , i ) given by
Eˆ k  t  
1
2
 d  aˆ
k
( )e  i t ,
(2)
where T is the arrival time difference for the signal and idler photons from the crystal to the
coupler. Then the two-photon coincidence detection rate is expressed as
S-1
R12   d G12(2) (t , t   ),
(3)
Tg
with the correlation function given by
G12(2) (t , t   )   Eˆ1† (t ) Eˆ 2† (t   ) Eˆ 2 (t   ) Eˆ1 (t )   0 Eˆ 2 (t   ) Eˆ1 (t ) 
2
,
(4)
where Tg represents the timing between the detection gates. Here we assume the pump light is an
ideal continuous-wave laser and thus neglect the average over the pump field. Substituting Eqs.
(1), (2) and the spontaneous parametric downconverted (SPDC) state into Eq. (4) we obtain
2
G12(2) (t , t   )  g (   T )  g (   T ) ,
(5)
where we define g (t )   ()eit d  and () denotes the spectrum amplitude. As Tg >> Tc ,
where Tc is the biphoton correlation time. The time integral range in Eq. (3) can be extended to
(-, +). Then after the time integral we obtain the coincidence rate
2
R12  1  Re   g  ( ) g (   T )d 


 1  Re    (  ) ( )e 2i T

 g ( ) d
d     ( )

2
(6)
d
For our source, the spectrum amplitude () has the following form
N
 ( ) 

N
f '()h() f (  m) 
m  N
rect( / B )sinc( A)
,
2
2
m  N (  )  (  m)

(7)
where f ()  sinc( A) is the phase matching spectrum function, with the full width at half
1, |  / B | 1/ 2
maximum (FWHM) as 2.78 / A, and h()  rect( / B)  
is the rectangular
0, |  / B | 1/ 2
spectrum function resulting from the FBG, with B denoting the width. The fiber Fabry-Perot
cavity
(FFPC)
has
a
Lorentzian
spectrum
bin
function
characterized
by:
f ( )  f s ( ) f i ( )  1 [(   i )(   i  )]  1 [(  ) 2   2 ] , where  is the spacing between
the frequency bins and 2 denotes the FWHM of each frequency bin. Corresponding to our
measurements, we evaluate our theory using the following parameter values: A = 2.78 / (2 
S-2
245 GHz) = 1.81 ps, B = 2  346 GHz = 2.2  1012 rad/s,  = 2  15.15 GHz = 95.2  1010
2
rad/s, and  = 2  1.36 GHz / 2 = 4.27  109 rad/s. The spectrum density  () can be
described by
N
2
2

() 
f '() h()
m  N
2
sinc2 ( A) rect( / B)
f (  m)  
,
2
2 2
m N [( )  (  m) ]
2
N
(8)
where we have neglected the overlaps among the frequency bins since  >> 2. Moreover, as
2
2
the FWHM of f '() , 2.78 / A is much larger than 2 and the width of h() , and B >>
2, we can make the following approximation
N
2
 () 

2
f '( Am) h(m)
2
f (  m)
2
m  N
sinc2 ( Am) rect( m / B) N0
sinc2 ( Am)
 

,

2
2 2
[(  ) 2  (  m) 2 ]2
m N
m  N0 [( )  (  m) ]
N
(9)
where N0  B / (2)  11 is the integer part of B / 2 . Thus we can obtain

N0
2

 ( ) d  
m  N0
  (  )  ( ) e
2 i  T
 d
sinc2 ( Am)


2
2 2
[( )  (  m ) ]
2( )3

sinc 2 ( Am).

m  N 0
2
 d
sinc 2 ( Am )
e 2 i  T
[(  ) 2  (  m ) 2 ]2
 e 2 | T | (1  2  |  T |) N0

sinc 2 ( A m) e 2 im T

3
2(  )
m  N 0

(10)
m  N0
d     (  ) e 2 i  T d 
N0

N0
(11)

 e 2 | T | (1  2  |  T |)  N 0
2
 2 sinc ( A m) cos(2 m T )  1 .
3
2(  )
 m 1

Then through Eq. (6) we arrive at the coincidence rate as
R12  1 

e2 |T | (1  2 |  T |)  N 0
2
 2 sinc ( Am) cos(2m T )  1 .
N0

 sinc2 ( Am)  m1
(12)
m   N0
S-3
We model the second-order correlation based on our experimental parameters. The
coincidence rate versus T in the range of {-320 ps, 320 ps} is plotted in Supplementary Figure
S1. We can see the interference fringe is a multi-dip pattern, with the dip revival period of about
33.2 ps, matching the 33.4 ps in our measurements. The linewidth of each Hong-Ou-Mandel dip
and the fall-off in visibility for increasing k bins from the zero delay point also matches with
the experimental observations.
Coincidence Rate, R12
1.0
0.8
0.6
0.4
0.2
0.0
-300
-200
-100
0
100
200
300
Relative Delay, T (ps)
Supplementary Figure S1 | Modeling of Hong-Ou-Mandel interference revivals for the
high-dimensional biphoton frequency comb. The coincidence counting rate R12 as a function
of T , the arrival time difference between the signal and idler photons. The fall-off in the
revived visibility away from the zero delay point arises from the Lorentzian lineshape of the
SPDC individual photons after passing through the cavity.
In the experiment, the bandwidth of FBG is more than that of the phase matching. It is
reasonable to assume an infinite number of frequency bins, i.e., N   , and making the
replacement of  T   T ' kT / 2 , where T  2 /  ,  A   T '  T / 2  A , and k is an integer
number, we can simplify Eq. (12) to
S-4
R12 ( T ' kT / 2) 
1  e2| T ' k / | (1  2 |  T ' k /  |)(1 |  T ' | / A)
 A   T '  A,

0
A   T '   /   A.

(13)
We can see the dip revival period is TR   /   33.2 ps (or T/2) and the visibility of the k-th
order dip is e2|k | /  (1  2 | k |  / ) . We note that the recurrence is at T/2 instead of at T in
our prior analysis where the beamsplitter was moved [J. H. Shapiro, Technical Digest of Topical
Conference on Nonlinear Optics, p.44, FC7-1, Optical Society of America (2002)]. With the
beamsplitter shift, the transmitted signal and idler beams do not experience an advance or a delay,
but one reflected beam is advanced while the other is delayed. In our case, only the idler beam is
delayed, giving rise to the T/2 recurrence in the coincidences as detailed above.
II. Theory of Franson interference of the high-dimensional biphoton frequency comb
The electric field operators at the two detectors D1 and D2 can be expressed as
1
Eˆ1  t    Eˆ s ( t )  Eˆ s ( t   T1 )  ,
2
1
Eˆ 2  t    Eˆ i ( t )  Eˆ i ( t   T2 )  ,
2
(14)
with the field detectors before the Franson interferometer Eˆk (t ) , ( k  s , i ) given by
Eˆ k  t  
1
2
 d  aˆ
k
(15)
( )e  i t ,
where T1 , T2 are the unbalanced arm differences. Then the two-photon coincidence detection
rate can be described by
R12   d G12(2) (t , t   ),
(16)
Tg
with the correlation function given by
G12(2) (t , t   )   Eˆ1† (t ) Eˆ 2† (t   ) Eˆ 2 (t   ) Eˆ1 (t )   0 Eˆ 2 (t   ) Eˆ1 (t ) 
2
,
(17)
where Tg represent the timing between the detection gates. Here we assume the pump light is an
ideal continuous-wave laser and thus neglect the average over the pump field. Substituting Eqs.
(1), (2) and the SPDC state into Eq. (4), we obtain
S-5
2
G12(2) (t , t   )  G (t , t )  G(t  T1 , t  T2 )  G(t  T1 , t )  G(t , t  T2 ) ,
(18)
with
G ( t1 , t 2 ) 
1  i p ( t1  t2  ) / 2
1  i p ( t1  t 2  ) / 2
e
 (  ) e i ( t2  t1  ) d  
e
g ( t 2  t1   ),

8
8
(19)
where () denotes the spectrum amplitude and we define g (t )   ()eit d  . Then we may
rewrite Eq. (5) as
G12(2) (t , t   )  e
e
 i p / 2
g ( )  e
i p ( T1 )/2
i p ( T1  T2 )/ 2
g (  T1 )  e
g (  T1  T2 )
 i p (  T2 )/ 2
2
(20)
g (  T2 ) .
As we have noted, T1 , T2 are much larger than the single-photon coherence time Tc ,
i.e., the range of the function g (t ) , so there is only one non-zero cross term in Eq. (20). Thus we
obtain
2
2
2
G1(2)
2 (t , t   )  g ( )  g (  T1  T2 )  g (  T1 )  g (  T2 )
2 Re[e
i p ( T1  T2 )/ 2
g  ( ) g (  T1  T2 )].
2
(21)
Since T1, T2 >> Tg, the coincidence detection system can resolve the short and long paths and
2
2
thus the two terms g (  T1 ) , g (  T2 ) have no contribution to the coincidence rate.
Moreover, as Tg >> Tc , the time integral range in Eq. (3) can be extended as (  ,  ) . Then
after the time integral we obtain the coincidence rate
R12 1 (T ) cos[[p T / 2  pT2  ],
(22)
where T  T1 T2 , and
( T )   g  ( ) g (  T ) d
   ()()eiT
2
 g ( ) d
d    ()
2
d    (T ) ei .
(23)
For our source, the spectrum amplitude () has the following form
S-6
N
N

 ( ) 
f '()h() f (  m) 
m  N
rect( / B )sinc( A)
,
2
2
m  N (  )  (  m)

(24)
2
Then we can write the spectrum density  () as
N
2
() 
2

f '() h()
m  N
2
sinc2 ( A) rect( / B)
f (  m)  
,
2
2 2
m N [( )  (  m) ]
2
N
(25)
where we have neglected the overlaps among the frequency bins since  >> 2. Moreover, as
2
2
the FWHM of f '() , 2.78 / A >> 2, and the width of h( ) , B >> 2 , we can make the
following approximation
2
N
 () 

2
f '( Am) h(m)
2
f (  m)
2
m  N
sinc2 ( Am) rect( m / B) N0
sinc2 ( Am)
 

,

2
2 2
[(  ) 2  (  m) 2 ]2
m N
m  N0 [( )  (  m) ]
N
(26)
where N0  B / (2)  11 is the integer part of B / (2) .Thus we can obtain

N0
2
 ( ) d  

m  N0
  (  )  ( ) e
iT
 d
sinc2 ( Am)


2
2 2
[( )  (  m ) ]
2( )3
N0

sinc 2 ( Am).
(27)
m  N0
2
d     ( ) e iT d 
N0


m  N 0
 d
sinc 2 ( A m)
eiT
[(  ) 2  (   m ) 2 ]2
 e   |T | (1   | T |) N0

sinc 2 ( A m) eimT

3
2(  )
m  N 0

(28)

 e   |T | (1   | T |)  N0
2
 2  sinc ( A m) cos( mT )  1 .
3
2(  )
 m 1

Then through Eqs. (6) and (10), we can write the coincidence rate as
R12  1 

e |T | (1   | T |)  N0
2
 2 sinc ( Am) cos(mT )  1
N0

 sinc2 ( Am)  m1
(29)
m  N 0
 cos[ p ( T / 2  T2 )].
S-7
Since p/2 = c/p = 1.43 × 1012 rad/s >> , , the term cos[p(T/2 + T2)] is the
fast collision part of the interference fringe with the other part determining the fringe envelope.
We can simulate our experimental results with the theoretical parameters above and for T2 = 5
ns.
If we consider a large number of frequency bins such as in our measurements, i.e.,
N   , and make the replacement of T  T ' kT , where -2A < T’  T - 2A, and k is an
integer number, we can simplify Eq. (16) to
R12 (  T ' kT ) 
1  e    | T '  2 k  /  | (1    |  T ' 2 k  /  |) [1 |  T ' | /(2 A ) ]

 cos[ p (  T '/ 2  k  /    T2 )]
 2 A   T '  2 A,


0
2 A   T '  2 /   2 A.

(30)
The fringe envelope of the coincidence rate versus T is plotted in Supplementary Figure S2. We
see that the interference fringe has a recurrent envelope that falls off away from the zero delay
point due to the Lorentzian lineshape of the FFPC. The recurrence period T is about 66 ps and
agrees well with repetition time of the biphoton frequency comb, i.e., the round trip time of the
FFPC.
The
maximum
visibility
at
the
k-th
order
peaks
can
be
found
to
be e2|k | /  (1  2 | k |  / ) . This gives an envelope visibility which matches well with the
experimental measurements.
S-8
Norm. coincidences R12
a
b
2.0
1.5
1.0
0.5
0.0
-300
Norm. coincidences R12
c
-200
-100
0
T (ps)
100
200
300
-0.5
0
T (ps)
0.5
2.0
1.5
1.0
0.5
0.0
-0.2
-0.1
0
T (ps)
0.1
0.2
Supplementary Figure S2 | Theory of Franson interference revival for the highdimensional biphoton frequency comb. a, The envelope of coincidence counting rates R12
plotted as a function of T. The interference has a recurrent envelope that falls off away from the
zero delay point, arising from the finite coherence time of the single frequency bin. b, zoom-in of
Franson interferences for the first time bin. c, Further zoom-in of the Franson interferences for
the first time bin. The high-frequency interference oscillations arise from the phase.
III. Characterization of Franson interferometer long-term stability
In our measurement, the fiber-based Franson interferometer needs to be stabilized at the
wavelength level over long term for the phase sensitive interference measurements. All
components are fixed on the aluminum housing with thermal conductive epoxy for good thermal
contact. Both interferometer arms are temperature-controlled with Peltier modules and sealed in
aluminum enclosures which themselves are also temperature stabilized. The delay line in arm2 is
based on a miniaturized linear stage with closed-loop piezoelectric motor control (CONEX-AGS-9
LS25-27P, Newport Corporation). The two fiber collimators are custom-built in-house with fine
focal adjustment for optimized pair performance and epoxied on the housing. The delay line is
aligned so that the double-pass insertion loss is less than 0.4  0.05 dB throughout the whole
travel range of 27 mm of the linear stage. The effective delay range is about 0 to 360 ps for the
reflected light, considering the double-pass optical path configuration and reflector setup.
Supplementary Figure S3 | Long-term stability test of the Franson interferometer. Arm1
and arm2 are connected in series for the classical interference test. Both interferometers are
double temperature controlled, and the delay line in arm2 is closed-loop piezoelectrically
controlled. Dual collimators are custom-built in-house with fine focal adjustment and optimized
performance. Fine-tuned alignment is such that the double-pass insertion loss is less than 0.4 dB
throughout the whole 27 mm travel range and up to 360 ps optical delay.
We have verified the stability of the Franson interferometer using classical interference
before the quantum correlation measurements. The setup is shown in Supplementary Figure S3.
The light is from a 1310 nm amplified spontaneous emission (ASE) source (S5FC1021S,
Thorlabs Inc.), and filtered with the same filter sub-assembly that is used in the biphoton
frequency comb generation. The two arms are connected in series so that classical interference
occurs between the events of passing long path of arm1, short path of arm2, and short path of
arm1, long path of arm2, while T1  T2 . The visibility of this interference is limited to 50%,
S-10
because of other events that contribute to the background. The intensity of the output is measured
using a photodiode (PDA20CS, Thorlabs Inc.), while tuning the temperature of arm1. The result
is shown in Supplementary Figure S4. The observed interferences agree well with a sinusoidal fit,
and the visibility is 49.8  1.0%, near or right at the classical limit over long measurement time
periods. This indicates the Franson interferometer is well stabilized. We can also obtain the delay
coefficient for this temperature tuning, which is observed to be 127 attoseconds per mK based on
the fringe period when the temperature is tuned.
Intensity (a.u.)
0.6
0.4
0.2
0.0
0
20
40
60
80
Temperature detuning (mK)
100
Supplementary Figure S4 | Classical interference visibility of the stabilized coupled
interferometers. Temperature of arm1 is tuned and the input is a 1310 nm ASE source. The
temperature-delay sensitivity is observed to be 127 attoseconds per mK.
IV. Stabilization of the pump laser in Franson interference
The Franson interference requires high stability – both short and long term – of the pump
laser, which defines the two-photon coherence time of the two-photon state. The coherence time
of the pump laser should be much longer than the path length difference. This path length
difference is 5 ns in our experiment. Therefore, we custom-built a stabilized laser at 658 nm
using self-injection-locking. The setup is shown in Supplementary Figure S5a, which is similar
to a Littman–Metcalf configuration external diode laser. The laser source that we use is a
standard Fabry-Perot laser diode with center wavelength around 658 nm (QLD-658-20S). It is
spatially single-mode, but has multiple longitudinal modes without feedback. A diffraction
S-11
grating is used for the longitudinal mode selection. The laser beam is first collimated onto the
grating. The first-order diffraction reflects off a tunable mirror back into the diode through the
grating. The zeroth order diffraction from the grating is used as the output. Because of the
grating, we succeed in achieving single-mode lasing, with a mode-rejection ratio over 30 dB.
The whole setup is isolated with double enclosures. The inner enclosure is made from aluminum
and temperature-stabilized. The outer laser housing is also a solid piece of aluminum that is
temperature-stabilized. A third temperature stabilization is applied to the diode.
Supplementary Figure S5 | Stabilized 658 nm pump laser with self-injection locking and
characterization through Franson-type interferometer. a, Layout schematic of custom-built
658 nm stabilized laser. 1, 658 nm Fabry-Perot laser diode with temperature stabilization; 2,
collimating lens; 3, diffraction grating; 4, high-reflection mirror; 5, temperature-stabilized laser
housing; 6, internal enclosure with temperature stabilization; 7, external enclosure. b, Franson
interference measurement to demonstrate pump laser stability, with the interferometer tuned to
the 0th time bin. Measurement is made at 6 mW pump power. The measured deviation of the
counting rate is about 5%, with the coincidence measurements taken every 40 seconds, which
corresponds to a 2 MHz drift of the pump laser. The long term drift, over 12 hours, is less than
100 MHz.
S-12
The current source for the diode is a low-noise laser diode controller (D2-105, Vescent
Photonics, Inc.). With these stabilizations, we achieve a free-running wavelength drift of the 658
nm laser at less than 2 MHz within 200 seconds, which is an integration time step for the
Franson measurement. The laser linewidth is measured classically with the Franson-type
interferometer setup. The pump power is about 6 mW in the measurement. We tune the
interferometer to the 0th time bin.
T
is set such that the coincidence counting rate is about the
middle of the sinusoidal fringe, which gives the best sensitivity to the pump drift. The
coincidence counting rate is taken every 40 seconds, and the result is shown in Supplementary
Figure S5b. The measured deviation of counting rate is about 5%, which corresponds to a pump
drift of 2 MHz. The long-term drift is less than 100 MHz within 12 hours (measured with a
wavelength meter, HighFinesse WS-7).
V. Measurements of Hong-Ou-Mandel revival of the high-dimensional biphoton frequency
comb
In the main text, we used an InGaAs/InP single-photon detector D2 with ~ 2.5 ns
effective gate width for the measurements, so that the detection gate widows of D1 and D2 are
always well-overlapped through the scanning range of relative delay.
Figure S6 illustrates another example of the Hong-Ou-Mandel revival when using an
effective detector gate width of  400 ps. The maximum visibility of the central dip can be
enhanced to 96.1%, because of the reduced accidental coincidence possibility. That visibility
becomes 97.8% after subtracting the accidental coincidence counts. We note that, in the
accidentals subtraction, the estimated double pairs are still included in the counts for the best
estimate of the visibility. The single-photon counting rate remains constant during the
measurement. However, here the background of the coincidence counting rate drops as the
relative delay increases (see Supplementary Figure S6a). This drop is from the temporal overlap
reduction for the gating of D1 and D2 at large relative delays of the HOM interferometer.
S-13
Supplementary Figure S6 | HOM measurement with single-photon detectors with an
effective gate width of  400 ps. a, Coincidence and single counting rates as a function of the
relative delay between the two arms of the HOM interferometer. The background of the
coincidence counting rate drops because of the reduced overlap between the detection windows
of D1 and D2. b, Zoom-in coincidence and single counting rate around zero relative delay
between the two arms. The visibility is measured to be 96.1 %, or 97.8 % after subtracting the
accidental coincidence counts.
VI. Polarization entanglement measurements of the high-dimensional biphoton frequency
comb
Before the hyperentanglement measurements, we test the polarization entanglement alone
for the hyperentangled state. Supplementary Figure S7a shows the experiment setup. We mix the
signal and idler photons on a 50:50 fiber coupler with orthogonal polarizations. By keeping the
relative delay  = 0, the signal and idler photons are well-overlapped temporally for the
interference. We present a measurement for the polarization entanglement by measuring the
coincidence counting rates while changing the angle of P2, when P1 was set at 45 o and 90 o,
respectively. As shown in Supplementary Figure S7b, both results fit well with sinusoidal curves,
with visibilities of 91.2  1.6% and 93.0  1.3%, which violate the Bell inequality by 12.8 and
17.7 standard deviations, respectively. This indicates the high-dimensional polarization
S-14
entangled state  
N

m  N
 p / 2  m
1
 p / 2  m 2  ( H
1
V
2
V
1
H 2 ) is generated with
high quality. Hence, in addition to the 4 frequency bits, the biphoton frequency comb also has
polarization entanglement for use as a high-dimensional quantum communications platform.
Supplementary Figure S7 | Polarization entanglement measurements of the highdimensional biphoton frequency comb. a, Illustrative experimental scheme. The signal and
idler photons are sent to a 50:50 fiber coupler with orthogonal polarizations for the generation of
polarization entanglement. P: polarizer; S.C.: single counts; C.C.: coincidence counts. b,
Polarization entanglement measurements with P1 fixed at 45o and 90o . In both cases, we
measured the coincidence counting rates at the two outputs while changing P2 from 0o to 360 o.
As shown by the black line (for P1 = 45o) and red line (for P1 = 90o), both measured results fit
well with sinusoidal curves, with accidentals-subtracted visibilities of 91.2 % and 93.0 %,
respectively.
VII. High-dimensional hyperentanglement and Bell inequality statistics
We performed a series of hyperentangled measurements of the biphoton frequency comb
with the visibilities summarized in Table 1 below, for the different P1 settings and different timebin settings, across the energy-time basis and the polarization basis. The resulting standard
deviation violation of the Bell inequality is computed correspondingly. Measurements are
performed at 80-second integration times based on the tradeoff between the setup’s long-term
S-15
stability over the complete hyperentanglement measurements and sufficiently reduced standard
deviations of the coincidence counts. The Clauser-Horne-Shimony-Holt (CHSH) S parameter is
determined from the polarization analysis angle set of the ( H V  V H ) triplet state:
S  C ( / 2, 7 / 8)  C ( / 4, 7 / 8)  C ( / 4, 5 / 8)  C ( / 2, 5 / 8)
(31)
Table 1 | Visibilities for the interference fringes in the high-dimensional hyperentanglement
measurement and Bell inequality violations. “st.d.” denotes standard deviation .
P1 = 45°,
Visibility V
Polarization
V (dark counts
subtracted)
Bell violation
(by st.d. )
basis
Time bin #0
Time bin #1
Time bin #2
Time bin #3
Time bin #4
82.9%
80.1%
77.3%
73.8%
77.3%
96.3%
96.2%
95.9%
93.8%
95.8%
8.83
9.46
10.5
6.8
9.33
P1 = 45°,
Visibility V
81.1%
74.1%
68.0%
60.0%
44.2%
Energy-time
V (dark counts
subtracted)
94.1%
87.5%
83.3%
74.7%
53.2%
Bell violation
(by st.d. )
7.34
5.19
4.83
2.12
none
P1 = 90°,
Visibility V
81.4%
76.6%%
75.8%
75.1%
75.3%
Polarization
96.9%
95.0%
95.8%
94.5%
95.7%
basis
V (dark counts
subtracted)
Bell violation
(by st.d. )
10.95
8.09
7.62
8.51
7.97
P1 = 90°,
Visibility V
80.3%
72.3%
67.5%
59.4%
44.2%
Energy-time
V (dark counts
subtracted)
Bell violation
(by st.d. )
95.9%
89.0%
84.3%
73.6%
54.1%
7.87
8.34
5.67
1.92
none
2.71
2.67
2.76
2.74
2.53
basis
basis
CHSH S parameter
S-16